Question 102
Let $a_{1}, a_{2}, a_{3}, \ldots$ be the sequence of positive integers such that
$$
a_{1}=1 \quad \text{ and } \quad a_{k+1}=a_{k}^{3}+1, \text{ for all positive integers } k
$$
Prove that for every prime number $p$ of the form $3 \ell+2$, where $\ell$ is a non-negative integer, there exists a positive integer $n$ such that $a_{n}$ is divisible by $p$.
$$
a_{1}=1 \quad \text{ and } \quad a_{k+1}=a_{k}^{3}+1, \text{ for all positive integers } k
$$
Prove that for every prime number $p$ of the form $3 \ell+2$, where $\ell$ is a non-negative integer, there exists a positive integer $n$ such that $a_{n}$ is divisible by $p$.